Gr obner Bases and Hypergeometric Functions - CiteSeerX

Grobner Bases and Hypergeometric Functions Bernd Sturmfels and Nobuki Takayama July 8, 1998

1 Introduction The purpose of this tutorial1 is to illustrate the use of Grobner bases and Buchberger's algorithm in the algebraic study of linear partial dierential equations. Our reference example is the following system of six dierential equations for a function (x1; x2; x3; x4) in four complex variables: @2 @x2 @x3 @3 @x2 @x24

@2 @x1 @x4 ; @ 3 3 ; @x3

= =

@ 3 @x21@x3 @ 3 @x1@x23

= =

@3 ; @x32 @3 @x22 @x4 ;

(1)

@ + x @ + x @ = a
 ; x1 @x@ + x2 @x 3 @x 4 @x x2 @x@ + 3x3 @x@ + 4x4 @x@ = b
 ; where a and b are complex parameters. Experts on hypergeometric functions will recognize these equations: this is the A-hypergeometric system of Gel'fand, Kapranov and Zelevinsky (1989) for the particular con guration  1   1   1   1  A = 0 ; 1 ; 3 ; 4 : 1

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Partially supported by the David and Lucile Packard Foundation Prepared for the book Grobner Bases and Applications, B. Buchberger and F. Winkler (eds.), Text and Monographs in Symbolic Computation, Springer Verlag, Vienna, 1998  1

1

We shall use Grobner bases to prove that, for every open ball U in C4, the dimension of the C-vector space of holomorphic functions  on U which satisfy (1) is at most ve. Moreover, for (a; b) = (1; 2) and suitable U it is exactly ve. See Propositions 2.1 and 4.1 for precise statements. One remark for experts: our example shows that Theorem 2 in (Gel'fand et al. 1989) is incorrect without an additional Cohen-Macaulayness assumption. The statement of this important theorem was corrected in (Gel'fand et al. 1993) but no explicit counterexample was given. This tutorial shows that it is easy to construct and study counterexamples by using Grobner bases. In the remainder of the introduction we motivate the equations (1) by describing some special solutions: polynomial solutions, algebraic solutions, rational solutions, and solutions admitting integral representations. In the sequel, any function  which satis es the dierential equations (1) is called A-hypergeometric. The relation to classical hypergeometric functions, mentioned in the tutorial by Chyzak (1998), can be seen in Theorem 4.1. We rst ask whether the system (1) has any polynomial solutions. A necessary condition for this is that (a; b) lies in the monoid NA spanned by the set A in N2, because the last two equations in (1) are equivalent to (sx1; stx2; st3x3; st4x4) = satb
(x1; x2; x3; x4): (2) To see this equivalence, apply the operators @=@sjs=1 and @=@tjt=1 to (2). Conversely, if (a; b) lies in NA then the following set is nite and non-empty F(a;b) = f(u1; u2; u3; u4) 2 N4 j u1 + u2 + u3 + u4 = a; u2 + 3u3 + 4u4 = bg; and there is exactly one A-hypergeometric polynomial up to scalar multiples: X xu1 xu2 xu3 xu4 ; (3) (a;b)(x) = u2F a;b u1! u2! u3 ! u4! 1

(

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)

which is a solution of (1). Next, consider the zeros of the following polynomial in one variable , f () = x1 + x2 + x33 + x44: They can be written in terms of radicals using Cardano's formula. These algebraic functions are A-hypergeometric for (a; b) = (0; ?1). In fact, if U is an open ball in C4 which does not intersect the zero set of the discriminant
A = x1x4
(27x21x43 +4x32x33 +6x4x1x22x23 +192x24x21x2x3 +27x24x42 ? 256x34x31); 2

then the four roots of f () = 0 are well-de ned holomorphic functions on U , and they form a C-basis for the space of A-hypergeometric functions on U . More generally, if m is a non-zero integer and (a; b) = (0; ?m) then a solution basis is given by the m-th powers of the roots 1; 2; 3; 4 of f . This observation leads to a nice rational A-hypergeometric function:

m1 + m2 + m3 + m4 : Indeed, the m-th power sum of the roots is a Laurent polynomial in x1; : : : ; x4 which satis es (1) with parameters (a; b) = (0; ?m). For instance, for m = 1 we get the A-hypergeometric function x3=x4, and for m = 6 we get

61 + 62 + 63 + 64 = (x63 ? 6x1x23x34 + 6x2x33x24 + 3x22x44) = x64: Further algebraic solutions to (1) for other integral parameter values can be found by considering residue integrals such as 1 Z zm dz; mi p = f 0(i) 2 ?1 Ci f (z) where Ci is a small, positively oriented loop around the point i in the complex plane. This is an algebraic function of x1; x2; x3; x4, and it satis es (1) for (a; b) = (?1; ?1 ? m). If we take a big loop C which positively encircles all four roots, then we get the rational A-hypergeometric function m1 + m2 + m3 + m4 : f 0(1) f 0(2) f 0(3) f 0(4) Finally, we mention a general integral representation for A-hypergeometric functions. If the complex parameters a and b are suciently generic, then every solution to (1) on an open ball U  C4 can be written as Z f (z)az?b?1 dz (4) (x1; x2; x3; x4) = ?

where ? is a segment with endpoints belonging to D = f0; 1; 2; 3; 4g. If the real parts of a and ?b ? 1 are less than or equal to ?1, then the integral does not converge. In this case, we take the nite part of the integral in the sense of Hadamard. How many segments are linearly independent? We 3

 assume that all the roots i are real numbers for simplicity. Among 52 = 10 segments, only four segments are independent; when all roots are ordered as  0 = 0 <  1 < 2 <  3 < 4 ; the integral (4) can be expressed as a C-linear combination of the four inteR grals [i;i ] f (z)az?b?1dz. The segments [i; i+1] are the compact chambers of a 1-dimensional hyperplane arrangement. They form a basis of the twisted homology group H1lf (CnD; r); see page 53 in (Aomoto and Kita 1994). Hypergeometric dierential equations play an important role in algebraic geometry because for certain rational parameters (a; b) the integral (4) can be regarded as periods of a family of curves; see (Yoshida 1997). +1

2 Characteristic variety and holonomicity

In this section we show how to prove that the space of A-hypergeometric functions is nite-dimensional, how to evaluate its dimension, and how to compute the characteristic variety. We use Grobner bases in the Weyl algebra

W = khx1; x2; x3; x4; @1; @2; @3; @4i over a eld k, which in our case is either Q or Q(a; b). The Weyl algebra W is the free associative algebra over k modulo the commutation rules xixj = xj xi; @i@j = @j @i; xi@j = @j xi for i 6= j ; and @ixi = xi@i + 1: A linear system of partial dierential equations with polynomial coecients corresponds to a left ideal in W . Our system (1) corresponds to the left ideal HA which is generated by (5) @2@3 ? @1@4; @12@3 ? @23; @2@42 ? @33; @1@32 ? @22@4; x1@1 + x2@2 + x3@3 + x4@4 ? a; x2@2 + 3x3@3 + 4x4@4 ? b: (6) Here a; b are either speci c numbers or indeterminate parameters. A function  on U  C4 is A-hypergeometric if it is annihilated by the left W -ideal HA. The theory (and practise) of Grobner bases works perfectly well for left ideals in the Weyl algebra W . We quickly review the relevant basics. Every element f in W can be written uniquely as a k-linear combination of normally 4

ordered monomials xa@ b = xa1 xa2 xa3 xa4 @1b @2b @3b @4b . This representation of f is called normally ordered representation. For example, the monomial @1x1@1 is not normally ordered. Its normally ordered representation is x1@12 + @1. Consider the commutative polynomial ring in eight variables 1

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gr (W ) = k[x1; x2; x3; x4; 1; 2; 3; 4] and the k-linear map gr : W ! gr (W ); xa@ b 7! xab . Let < be any term order on gr (W ). This gives a total order among normally ordered monomials in W via xA@ B > xa@ b , xAB > xab . For any element f 2 W let in< (f ) denote the highest monomial xA @ B in the normally ordered representation of f . If I is a left ideal in W then its initial ideal is the ideal gr (in< (I )) in gr (W ) generated by all monomials gr (in< (f )) for f 2 I . Clearly, gr (in< (I )) is generated by nitely many monomials xab . A nite subset G of I is called a Grobner basis of I with respect to the term order < if fgr (in< (g )) j g 2 Gg generates gr (in